No description available.

No description available.

[pt] A idéia central desta tese foi trabalhar o conceito de fração, identificando a fração como número e representando esse número na reta numérica, tendo como base as recomendações e experiências realizadas por Kathleen Hart, e principalmente Hung-Hsi Wu. Um conjunto de atividades foi desenvolvido, constituindo uma proposta de ensino de frações, posta em prática em turmas do sétimo ano e oitavo ano do Colégio Pedro II durante os anos letivos de 2006 e 2007. Parte substancial da tese é dedicada à descrição dos procedimentos metodológicos desenvolvidos para a aplicação dos princípios teóricos defendidos e de aspectos da implementação desses procedimentos em sala de aula. Grande número de exemplos das atividades realizadas é apresentado, dando-se destaque às resposta dos alunos às mesmas. O acompanhamento da evolução dos alunos é amplamente documentado. Os resultados são avaliados tanto do ponto do vista global quanto do ponto de vista de grupos diferenciados de alunos em diferentes níveis de qualificação prévia. O encaminhar de todo o trabalho investigativo leva a duas conclusões essenciais: a familiarização com o campo algébrico é alcançada com mais facilidade quando o conceito de fração é trabalhado na forma proposta; o envolvimento com as atividades e a compreensão pelos alunos da construção que estavam realizando em relação aos conceitos propostos devolvem o interesse das crianças pela aprendizagem da Matemática, fundamental para o êxito de qualquer proposta pedagógica. / [en] The main idea of this thesis was to work on the concept of fraction, identifying the fraction as a number and representing it in the line, taking as basis the recommendations and experiments of Kathleen Hart, and, mainly Hung-Hsi Wu. A set of activities was developed, generating a proposal for teaching fractions, led to practice in classes of students of 7th and 8th grade of Colegio Pedro II during the school years of 2006 and 2007. A substantial part of the thesis is devoted to the description of the methodological procedures developed for the implementation of the theoretical principles defended and of aspects of the implementation of these procedures in the classroom. Large number of examples is given of the activities performed, emphasis being given to the response of the students. The monitoring of the progress of the students is widely documented. The results are evaluated from the global point of view and considering distinct groups of students at different levels of prior qualification. The development of all the research work led essentially to two conclusions: the familiarization with the algebraic field is more easily reached when the concept of fraction is dealt in the proposed way; the involvement of the students with the activities and their understanding of the construction that they were doing with the concepts proposed returned them the interest for learning mathematics, the key to the success of any educational proposal.

Submitted by Augusto Sérgio Furquim null (serginspfc@gmail.com) on 2018-02-28T01:21:15Z No. of bitstreams: 1 Dissertacao_Augusto_ProfMat_Final.pdf: 4745307 bytes, checksum: 2d60b356f245a77096b1ce3cb1292751 (MD5) / Rejected by Elza Mitiko Sato null (elzasato@ibilce.unesp.br), reason: Solicitamos que realize correções na submissão seguindo as orientações abaixo: Problema 01) Falta a FOLHA DE APROVAÇÃO (Obrigatório pela ABNT NBR14724) Na página da Seção de pós-graduação, em Instruções para Qualificação e Defesas de Dissertação e Tese, você pode acessar o modelo das páginas pré-textuais. Lembramos que o arquivo depositado no repositório deve ser igual ao impresso. Agradecemos a compreensão on 2018-02-28T17:41:39Z (GMT) / Submitted by Augusto Sérgio Furquim null (serginspfc@gmail.com) on 2018-03-01T02:15:50Z No. of bitstreams: 1 Dissertacao_Augusto_ProfMat_Final.pdf: 4746758 bytes, checksum: 726978107a9190735785165300342924 (MD5) / Approved for entry into archive by Elza Mitiko Sato null (elzasato@ibilce.unesp.br) on 2018-03-01T15:28:52Z (GMT) No. of bitstreams: 1 furquim_as_me_sjrp.pdf: 4746758 bytes, checksum: 726978107a9190735785165300342924 (MD5) / Made available in DSpace on 2018-03-01T15:28:52Z (GMT). No. of bitstreams: 1 furquim_as_me_sjrp.pdf: 4746758 bytes, checksum: 726978107a9190735785165300342924 (MD5) Previous issue date: 2018-02-19 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Neste trabalho, promovemos um estudo acerca da introdução `a Álgebra nos Anos Finais do Ensino Fundamental. Para tanto, perpassamos pela construção lógica e a Aritmética do conjunto dos números Naturais, Inteiros e Racionais. Além disso, abordamos a Álgebra em seus aspectos históricos, concepções, resultados em avaliações de larga escala, disposição no Currículo do Estado de São Paulo - onde constatamos um salto entre Aritmética e Álgebra - e sua introdução através de situações de aprendizagem as quais denominamos atividades de “Pré-Álgebra”. Ao longo do desenvolvimento de nosso trabalho discutimos, em revisão de literatura e, também, experimentalmente, as implicações de um adiantamento, com relação ao currículo supracitado, no tratamento da Álgebra em atividades introdutórias. Como resultado, verificamos que há, sim, espaço para que a Álgebra seja previamente introduzida, em consonância com a Aritmética, através de atividades de Pré-Álgebra. No entanto, não podemos garantir que esta introdução seja mais fácil para os alunos, embora conjecturemos que a suavidade da forma como tratamos a Álgebra durante as atividades da Pré-Álgebra pode contribuir para uma melhoria no cenário de ensino da Álgebra em nosso país. / In this work, we promote a study on the introduction to Algebra in the final years of Elementary School. For this, we went through the logical construction and the arithmetic of the set of Natural, Integer and Rational numbers. Moreover, we analyzed Algebra based on its historical aspects, conceptions, the outcome on the large-scale evaluations, on how it appears on the State of S˜ao Paulo Curriculum - which showed us a jump between Arithmetic and Algebra - and its introduction by using some learning activities that we call ”pre-algebra”. Throughout the development of our work we discussed - by reviewing some literature and also experimentally - the implications of an advance in the treatment of Algebra in introductory activities, in comparison with the above mentioned curriculum. As a result, we find out that in fact there is space to introduce Algebra in advance, in line with Arithmetic, using the Pre-Algebra activities. However, we cannot guarantee that this introduction is going to be easier for the students, although we conjecture that the smoothness of the way we treat Algebra during the PreAlgebra activities can contribute for an improvement in the scenario of teaching Algebra in our country.

Vi har funderat över hur vi som blivande lärare skulle kunna förbättra elevernas förståelse inom algebra. Syftet med detta arbete var att ta reda på vad symbolen har för betydelse vid algebraräkning och huruvida det kan vara givande att introducera pre-algebra i större utsträckning i tidigare årskurs. Med symbol menar vi det tecken som representerar den okända variabeln i algebra. För att undersöka detta har en observationsstudie genomförts i årskurs fem och två tester i årskurs två och sju. Sammanlagt deltog 178 elever. Undersökningarna visade att valet av symbol har viss betydelse, men är dock ej helt avgörande, i synnerhet i grundskolans tidiga år. Vidare visade studien att barn i årskurs två är mogna för pre-algebra, vilket betyder att den kan introduceras i större utsträckning redan då. För att underlätta övergången från pre-algebra till algebra kan läraren variera valet av symbol. Förslagsvis en symbol det går att skriva inuti till att börja med, för att sedan successivt skifta till en symbol som ej går att skriva i.

No description available.

­­­­­­­­­­I flera decennier har likhetstecknets betydande roll i matematikämnet studerats av en rad olika forskare. Där det tydligt framgår att den operativa uppfattningen av likhetstecknet dominerar i klassrummen, därav syftar vår studie till att undersöka hur elevers uppfattning och förståelse av likhetstecknets relationella funktion förändras om olika delar av undervisningsinnehållet varieras. Syftet kommer besvaras genom frågeställningarna; Hur visar sig elevernas förståelse för likhetstecknets funktion i förtestet, hur påverkar variationen av undervisningsinnehållet elevernas förståelse av likhetstecknet och vilken uppfattning av likhetstecknet visas av eleverna i eftertestet efter en variation av undervisningsinnehållet. Studien har sin utgångspunkt i variationsteorin som fokuserar på vad som ska läras och hur lärandet kan möjliggöras genom teorins centrala begrepp, kritiska aspekter och variationsmönster. För att uppfylla studiens syfte och besvara frågeställningarna kommer designbaserad forskning användas som forskningsmetod, i form av en lektionsserie på tre lektioner som baserats på ett förtest.  Resultatet visar att variation av undervisningsinnehållet kan med hjälp av variationsmönster; kontrastering, generalisering och fusion, synliggöra och urskilja de kritiska aspekterna för att kunna utveckla elevernas förståelse av likhetstecknet. / For several decades, the significant role of the equals sign in the subject of mathematics has been studied by several different researchers, where the operative conception of the equal sign dominates in the classroom. Therefore, our study aims to investigate how students' conception and understanding of the equal sign relational function changes if different parts of the teaching content are varied. In accordance with this aim, the following research questions are addressed in the present study; What is the students' understanding of the function of the equal sign in the pre-test, how does the variation of the teaching content affect the students' understanding of the equal sign and what perception of the equal sign is shown by the students in the post-test after a variation of the teaching content. The study has its starting point in variation theory, which focuses on what is to be learned and how learning can be made possible through the theory's central concepts, critical aspects, and patterns of variation. To fulfil the purpose of the study and answer the questions, design-based research will be used as a research method, in the form of a series of lessons (3) based on a pre-test.  The results show variation of the teaching content can be by means of patterns of variation; contrasting, generalizing, and fusion, making visible and distinguishing the critical aspects to develop the students' understanding of the equal sign.

Från och med höstterminen 2018 är programmering en del av kursplanen i matematik, och har då placerats under det centrala innehållet för algebra. Trots att det inte är obligatoriskt förrän höstterminen har lärare redan börjat undervisa i programmering, och appar är då en populär lärresurs. Hur väl stämmer apparna för programmering överens med kursplanen i matematik, och hur väl lämpar de sig för undervisning på lågstadiet? Med utgångspunkt i dessa frågor gjordes en analys av 14 programmeringsappar som riktar sig till lågstadieåldrarna, 6-10 år. Studien är en kvalitativ innehållsanalys med en analysmodell som baserats på kursplanen i matematik, teorier om pre-algebra samt kriterier som lärare anser viktiga för en bra app. Studien visar att apparnas lämplighet för matematikundervisning varierar, men att det är de appar som är utformade som spel som stämmer bäst överens med kursplanen i matematik. Det är också dessa appar som har en pedagogisk utformning som lämpar sig bäst för undervisning på lågstadiet. Resultatet av studien ifrågasätter dock valet att placera programmering under algebra i läroplanen, då kopplingen däremellan inte är helt tydlig. / Starting in the fall of 2018, programming is a part of the Swedish curriculum for mathematics and has been placed under the core content for algebra. Even though it is not obligatory until the fall, some teachers have already started teaching about programming using apps as a digital learning resource. How well do the programming apps fit with the curriculum for mathematics, and how suitable are they to be used in primary school education? With these questions in mind an analysis was made of 14 programming apps for children 6-10 years old. The study is a qualitative content analysis with an analysis model based on the curriculum for mathematics, theories about pre-algebra and criteria that teachers believe to be important in the choice of a good app. The study shows that the apps’ suitability for mathematics education vary, but it is the apps that are designed as games that fit best with the curriculum for mathematics. These apps also have the pedagogical design that suits best for primary school education. The study’s results question the choice of placing programming under algebra in the curriculum, since the connection between the areas is not quite clear.

This study collected evidence to address the assumptions underlying the use of the Utah Core Assessment to Pre-Algebra (UCAP) to (a) measure student achievement in pre-algebra, and (b) assist teachers in making adjustments to instruction. An evaluative argument was defined to guide the collection of evidence. Each of the assumptions in the evaluative argument was addressed using data from a suburban northern Utah school district. To collect the evidence, test content was examined including item match to course objectives, reliability, and subtest intercorrelations. Analyses of correlations of the UCAP with convergent and discriminant measures were completed using student test data (N = 1,461), including an examination of both the pattern of correlations and tests of statistical significance. Pre-algebra teachers (N = 12) were interviewed to ascertain the degree to which UCAP results were used to make necessary adjustments to instruction. It was found that the UCAP was technically sound, but measured only 65% of course objectives. Correlation coefficients were analyzed using pattern comparisons and tests of statistical significance. It was found that the pattern of correlation coefficients and the distinction of convergent and discriminant measures supported the UCAP as a measure of mathematics. Teacher interview data revealed that teachers did not make substantive adjustments to the instruction of pre-algebra based on test scores. Based on these results it was concluded that the underlying assumptions concerning the use of the UCAP were not fully supported. The lack of complete coverage of the pre-algebra course objectives calls into question the ability of the UCAP scores to be used as measures of student achievement, in spite of the technical quality of the test. There was support for the assumption that the UCAP measures mathematics. There was little evidence that teachers use the UCAP score reports to make meaningful and appropriate adjustments to instruction. More evidence is needed to understand the factors that may have led to this lack of use. The evaluative argument framework defined in this study provides guidance for future research to collect evidence of the validity of decisions based on UCAP scores.

Detta är en empirisk studie som har syftat till att synliggöra vilka strategier elever i årskurs två har använt när de löst öppna utsagor samt vilka missuppfattningar som förekommit i samband med detta. Studien har även syftat till att synliggöra hur eleverna har kunnat relatera öppna utsagor till vardagliga sammanhang. Vid datainsamlingen, som utfördes på två olika skolor, genomfördes 15 kvalitativa intervjuer vilka kombinerades med observationer. Eftersom det i uppsatsens syfte finns en underförstådd relation till fenomenografin och hermeneutiken i form av såväl teorier som metodologier har dessa forskningsansatser varit en stor inspirationskälla i vår undersökning. Vidare har vi genom vår studie visat att de strategier som elever använder när de löser slutna utsagor till stora delar även används när eleverna löser uppgifter i form av öppna utsagor. Resultatet har visat på en bred variation i elevernas lösningsstrategier och det framgår att eleverna, vid deras lösningar av öppna subtraktionsutsagor, knappt använder de tre vanligaste subtraktionsstrategierna. Istället använder de bland annat en strategi som ger upphov till flera missuppfattningar. Det visade sig även att samtliga elever hade svårt att relatera öppna utsagor till vardagliga sammanhang.